In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.
High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of high-order derivatives that do not occur in the methodology. These results cannot, therefore, ensure that the method converges in cases where there are no such high-order derivatives. However, the method could converge. In this paper, we are developing a process in which both the local and semi-local convergence analyses of two related methods of the sixth order are obtained exclusively from information provided by the operators in the method. Numeric applications supplement the theory.
Symmetries are vital in the study of physical phenomena such as quantum physics and the micro-world, among others. Then, these phenomena reduce to solving nonlinear equations in abstract spaces. These equations in turn are mostly solved iteratively. That is why the objective of this paper was to obtain a uniform way to study three-step iterative methods to solve equations defined on Banach spaces. The convergence is established by using information appearing in these methods. This is in contrast to earlier works which relied on derivatives of the higher order to establish the convergence. The numerical example completes this paper.
Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.
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